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arXiv · 2412.10971

Soliton solutions associated with a class of third-order ordinary linear differential operators

Tuncay Aktosun·Abdon E. Choque-Rivero·Ivan Toledo·Mehmet Unlu
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Abstract

Explicit solutions to the related integrable nonlinear evolution equations are constructed by solving the inverse scattering problem in the reflectionless case for the third-order differential equation $d^3ψ/dx^3+Q\,dψ/dx+Pψ=k^3ψ,$ where $Q$ and $P$ are the potentials in the Schwartz class and $k^3$ is the spectral parameter. The input data set used to solve the relevant inverse problem consists of the bound-state poles of a transmission coefficient and the corresponding bound-state dependency constants. Using the time-evolved dependency constants, explicit solutions to the related integrable evolution equations are obtained. In the special cases of the Sawada--Kotera equation and the modified bad Boussinesq equation, the method presented here explains the physical origin of the constants appearing in the relevant $\mathbf N$-soliton solutions algebraically constructed, but without any physical insight, by the bilinear method of Hirota.

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Tuncay Aktosun, Abdon E. Choque-Rivero, Ivan Toledo, Mehmet Unlu. 2025-04-29. Soliton solutions associated with a class of third-order ordinary linear differential operators. https://arxiv.org/abs/2412.10971

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